Some improvements for the algorithm of Gröbner bases over dual valuation domain

As a special ring with zero divisors, the dual noetherian valuation domain has attracted much attention from scholars. This article aims at to improve the Buchberger's algorithm over the dual noetherian valuation domain. We present some criterions that can be applied in the algorithm for computing Gröbner bases, and the criterions may drastically reduce the number of S-polynomials in the course of the algorithm. In addition, we clearly demonstrate the improvement with an example

Bivariate Polynomial Matrix and Smith Form

Matrix equivalence plays a pivotal role in multidimensional systems, which are typically represented by multivariate polynomial matrices. The Smith form of matrices is one of the important research topics in polynomial matrices. This article mainly investigates the Smith forms of several types of bivariate polynomial matrices and has successfully derived several necessary and sufficient conditions for matrix equivalence

Generalized Serre Problem over Elementary Divisor Rings

Matrix factorization has been widely investigated in the past years due to its fundamental importance in several areas of engineering. This paper investigates completion and zero prime factorization of matrices over elementary divisor rings (EDR). The Serre problem and Lin-Bose problems are generalized to EDR and are completely solved

On Serre Reduction of Multidimensional Systems

Serre reduction of a system plays a key role in the theory of Multidimensional systems, which has a close connection with Serre reduction of polynomial matrices. In this paper, we investigate the Serre reduction problem for two kinds of nD polynomial matrices. Some new necessary and sufficient conditions about reducing these matrices to their Smith normal forms are obtained. These conditions can be easily checked by existing Gröbner basis algorithms of polynomial ideals

Minor Prime Factorization for n-D Polynomial Matrices over Arbitrary Coefficient Field

In this paper, we investigate two classes of multivariate (n-D) polynomial matrices whose coefficient field is arbitrary and the greatest common divisor of maximal order minors satisfy certain condition. Two tractable criterions are presented for the existence of minor prime factorization, which can be realized by programming and complexity computations. On the theory and application, we shall obtain some new and interesting results, giving some constructive computational methods for carrying out the minor prime factorization